57.8.1 problem 1

Internal problem ID [14407]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.2.4. Applications. Exercises page 99
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:36:48 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+x^{\prime }+x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ x^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.095 (sec). Leaf size: 28

ode:=diff(diff(x(t),t),t)+diff(x(t),t)+x(t) = 0; 
ic:=[x(0) = 1, D(x)(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = {\mathrm e}^{-\frac {t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\cos \left (\frac {\sqrt {3}\, t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 42

ode=D[x[t],{t,2}]+D[x[t],t]+x[t]==0; 
ic={x[0]==1,Derivative[1][x][0 ]==1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t/2} \left (\sqrt {3} \sin \left (\frac {\sqrt {3} t}{2}\right )+\cos \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t) + Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 1} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\sqrt {3} \sin {\left (\frac {\sqrt {3} t}{2} \right )} + \cos {\left (\frac {\sqrt {3} t}{2} \right )}\right ) e^{- \frac {t}{2}} \]